poles and zeros

Poles and zeros are the special points in the complex s-plane that completely characterise a linear system's transfer function. Poles are the values of s where G(s) blows up to infinity (the roots of the denominator); zeros are where G(s) drops to zero (the roots of the numerator). The poles are the more important half: each pole is a natural mode the system wants to move in — like the resonant frequencies of a bell, except a pole also encodes whether that mode decays, holds, or grows.

The location of a pole on the plane is destiny. A pole in the left half-plane (negative real part) corresponds to a mode that decays — stable. A pole in the right half-plane grows without bound — unstable. A pole on the imaginary axis rings forever. The distance from the imaginary axis sets how fast it settles; the angle off the real axis sets how much it overshoots and oscillates. Zeros don't cause instability but they sculpt the response: a zero in the right half-plane gives the notorious 'wrong-way' dip, where the output first lurches the opposite direction (think a car backing up slightly before lurching forward). Read the pole-zero map and you've read the system's soul.